In this report I will go through the use of Modern Portfolio Theory to optimize a portfolio of any number of stock holdings by measuring the annual returns against the annualized risk, while recording the Sharpe Ratio of each portfolio.
Select tickers that make up your portfolio and assign the price data.
tick <- c('SCHG','RSP','SPLG','IHI','TSM',
'MSFT', 'EFX','FCX','BBJP', 'XLRE', 'DIS', 'PYPL',
'AMZN', 'PSA', 'GOOG', 'VPU')
price_data <- tq_get(tick, from = '2020-01-01',
get = 'stock.prices')
## change the from date if you want further historical data, I like to do
## around 10 years
## enter the weights of your portfolio make sure the weights correspond to the
## order of tickers entered in tick above
my_wts <- c(0.1463, 0.1269, 0.0661, 0.0646, 0.0518, 0.0456,
0.0408, 0.0407, 0.0397, 0.0387, 0.0358,
0.0345, 0.0332, 0.0328, 0.0307, 0.0261)Next transform the data using log scale
log_ret_tidy <- price_data %>%
group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = 'daily',
col_rename = 'ret',
type = 'log')
head(log_ret_tidy)## # A tibble: 6 × 3
## # Groups: symbol [1]
## symbol date ret
## <chr> <date> <dbl>
## 1 SCHG 2020-01-02 0
## 2 SCHG 2020-01-03 -0.00554
## 3 SCHG 2020-01-06 0.00628
## 4 SCHG 2020-01-07 -0.00138
## 5 SCHG 2020-01-08 0.00699
## 6 SCHG 2020-01-09 0.00914log_ret_xts <- log_ret_tidy %>%
spread(symbol, value = ret) %>%
tk_xts()## Warning: Non-numeric columns being dropped: date## Using column `date` for date_var.head(log_ret_xts)## AMZN BBJP DIS EFX FCX
## 2020-01-02 0.000000000 0.0000000000 0.0000000000 0.000000000 0.000000000
## 2020-01-03 -0.012213314 -0.0105434629 -0.0115372638 0.003365160 -0.030771493
## 2020-01-06 0.014775871 0.0036622412 -0.0058189870 0.004260731 0.003898638
## 2020-01-07 0.002089436 0.0016233857 0.0003432508 -0.001883466 0.015444163
## 2020-01-08 -0.007839287 0.0004054758 -0.0020611691 0.021210391 0.009912407
## 2020-01-09 0.004787695 0.0060617598 -0.0039278697 0.011554637 -0.016832816
## GOOG IHI MSFT PSA PYPL
## 2020-01-02 0.0000000000 0.000000000 0.000000000 0.0000000000 0.000000000
## 2020-01-03 -0.0049193493 -0.007216658 -0.012529993 0.0107264290 -0.018131770
## 2020-01-06 0.0243581503 0.008345515 0.002581339 0.0035500870 0.012880971
## 2020-01-07 -0.0006242443 -0.002032792 -0.009159304 -0.0039712338 -0.004548771
## 2020-01-08 0.0078495086 0.004211534 0.015802594 0.0018241000 0.019414593
## 2020-01-09 0.0109839096 0.008816560 0.012415846 0.0004203794 0.006684815
## RSP SCHG SPLG TSM VPU
## 2020-01-02 0.0000000000 0.000000000 0.000000000 0.000000000 0.0000000000
## 2020-01-03 -0.0057035680 -0.005538329 -0.006312541 -0.033533995 0.0002833025
## 2020-01-06 0.0012991377 0.006281609 0.003161252 -0.011606873 0.0019817689
## 2020-01-07 -0.0008657874 -0.001380641 -0.002633594 0.016074890 -0.0025484543
## 2020-01-08 0.0029408196 0.006990197 0.005260586 0.007346080 0.0000000000
## 2020-01-09 0.0045672302 0.009140058 0.007318301 0.008137101 0.0045260057
## XLRE
## 2020-01-02 0.0000000000
## 2020-01-03 0.0072987978
## 2020-01-06 0.0002599757
## 2020-01-07 -0.0109661283
## 2020-01-08 0.0049755020
## 2020-01-09 -0.0002612056tail(log_ret_xts)## AMZN BBJP DIS EFX FCX
## 2023-05-25 -0.015102768 0.005059203 -0.010496122 -0.0029406852 0.012567550
## 2023-05-26 0.043475866 0.003224506 0.001700409 0.0168038732 0.033336402
## 2023-05-30 0.012822304 -0.006256986 -0.005337600 0.0005695070 -0.015652520
## 2023-05-31 -0.008916851 -0.008948578 0.001592894 -0.0102056274 0.003208421
## 2023-06-01 0.017999211 0.023820268 0.007136788 -0.0003355566 0.032376443
## 2023-06-02 0.011983006 0.019947305 0.024309854 0.0241602982 0.047352568
## GOOG IHI MSFT PSA PYPL
## 2023-05-25 0.022034300 -0.0088155387 0.037736809 -0.0009886295 -0.0401143218
## 2023-05-26 0.008647678 -0.0005653933 0.021160155 0.0125660382 0.0142155130
## 2023-05-30 -0.006318259 -0.0069989680 -0.005059560 0.0076790852 0.0299359560
## 2023-05-31 -0.010241585 0.0043564714 -0.008550618 -0.0195742226 -0.0009673906
## 2023-06-01 0.008073023 0.0118367012 0.012678421 0.0004587895 0.0169549415
## 2023-06-02 0.006891058 0.0091103701 0.008443439 0.0173486283 0.0143298233
## RSP SCHG SPLG TSM VPU
## 2023-05-25 -0.0006429918 0.020643492 0.0084440798 0.113372272 -0.013992142
## 2023-05-26 0.0083271410 0.019941832 0.0132425596 0.022140424 0.000569165
## 2023-05-30 -0.0017734904 0.002695155 -0.0004048676 -0.011988990 -0.003776673
## 2023-05-31 -0.0087716429 -0.004401194 -0.0050746075 -0.033806949 0.008884524
## 2023-06-01 0.0074928315 0.013567182 0.0089142089 0.002532545 -0.006602615
## 2023-06-02 0.0215908529 0.009639000 0.0150137936 0.001011286 0.011050602
## XLRE
## 2023-05-25 0.0022695565
## 2023-05-26 0.0115510465
## 2023-05-30 0.0030765122
## 2023-05-31 0.0064022358
## 2023-06-01 0.0002773844
## 2023-06-02 0.0208627421Next calculate the mean returns, and covariance matrix
mean_ret <- colMeans(log_ret_xts, na.rm = TRUE)
print(round(mean_ret, 5))## AMZN BBJP DIS EFX FCX GOOG IHI MSFT
## 0.00031 0.00010 -0.00057 0.00050 0.00124 0.00070 0.00024 0.00089
## PSA PYPL RSP SCHG SPLG TSM VPU XLRE
## 0.00052 -0.00064 0.00032 0.00051 0.00038 0.00066 0.00012 0.00008cov_mat <- cov(log_ret_xts, use= 'complete') * 252
print(round(cov_mat,4))## AMZN BBJP DIS EFX FCX GOOG IHI MSFT PSA PYPL
## AMZN 0.1505 0.0363 0.0663 0.0614 0.0770 0.0907 0.0511 0.0921 0.0344 0.1106
## BBJP 0.0363 0.0395 0.0431 0.0384 0.0678 0.0405 0.0333 0.0411 0.0234 0.0467
## DIS 0.0663 0.0431 0.1393 0.0650 0.1175 0.0694 0.0539 0.0679 0.0382 0.0915
## EFX 0.0614 0.0384 0.0650 0.1292 0.0797 0.0641 0.0565 0.0675 0.0488 0.0861
## FCX 0.0770 0.0678 0.1175 0.0797 0.3267 0.0901 0.0763 0.0884 0.0489 0.1153
## GOOG 0.0907 0.0405 0.0694 0.0641 0.0901 0.1185 0.0575 0.0945 0.0386 0.0966
## IHI 0.0511 0.0333 0.0539 0.0565 0.0763 0.0575 0.0627 0.0617 0.0388 0.0755
## MSFT 0.0921 0.0411 0.0679 0.0675 0.0884 0.0945 0.0617 0.1164 0.0458 0.1036
## PSA 0.0344 0.0234 0.0382 0.0488 0.0489 0.0386 0.0388 0.0458 0.0816 0.0448
## PYPL 0.1106 0.0467 0.0915 0.0861 0.1153 0.0966 0.0755 0.1036 0.0448 0.2440
## RSP 0.0498 0.0402 0.0702 0.0617 0.1027 0.0596 0.0530 0.0613 0.0425 0.0719
## SCHG 0.0852 0.0413 0.0698 0.0675 0.0952 0.0833 0.0605 0.0886 0.0433 0.1041
## SPLG 0.0613 0.0379 0.0646 0.0590 0.0913 0.0660 0.0528 0.0699 0.0408 0.0790
## TSM 0.0725 0.0446 0.0617 0.0565 0.1147 0.0744 0.0515 0.0772 0.0284 0.0939
## VPU 0.0310 0.0293 0.0453 0.0511 0.0582 0.0407 0.0444 0.0461 0.0488 0.0455
## XLRE 0.0483 0.0365 0.0589 0.0642 0.0805 0.0552 0.0542 0.0613 0.0626 0.0669
## RSP SCHG SPLG TSM VPU XLRE
## AMZN 0.0498 0.0852 0.0613 0.0725 0.0310 0.0483
## BBJP 0.0402 0.0413 0.0379 0.0446 0.0293 0.0365
## DIS 0.0702 0.0698 0.0646 0.0617 0.0453 0.0589
## EFX 0.0617 0.0675 0.0590 0.0565 0.0511 0.0642
## FCX 0.1027 0.0952 0.0913 0.1147 0.0582 0.0805
## GOOG 0.0596 0.0833 0.0660 0.0744 0.0407 0.0552
## IHI 0.0530 0.0605 0.0528 0.0515 0.0444 0.0542
## MSFT 0.0613 0.0886 0.0699 0.0772 0.0461 0.0613
## PSA 0.0425 0.0433 0.0408 0.0284 0.0488 0.0626
## PYPL 0.0719 0.1041 0.0790 0.0939 0.0455 0.0669
## RSP 0.0661 0.0625 0.0600 0.0591 0.0504 0.0620
## SCHG 0.0625 0.0831 0.0668 0.0757 0.0450 0.0606
## SPLG 0.0600 0.0668 0.0595 0.0615 0.0463 0.0574
## TSM 0.0591 0.0757 0.0615 0.1553 0.0313 0.0506
## VPU 0.0504 0.0450 0.0463 0.0313 0.0669 0.0597
## XLRE 0.0620 0.0606 0.0574 0.0506 0.0597 0.0794Now we will find some random weights for the holdings in the portfolio, as well as finding that weighted portfolio’s returns, risk, and Sharpe ratio.
wts <- runif(n = length(tick))
wts <- wts/sum(wts)
print(wts)## [1] 0.05115565 0.01187139 0.04695456 0.09506853 0.13586896 0.01062504
## [7] 0.10156061 0.10008125 0.03874295 0.04424691 0.01679664 0.02361168
## [13] 0.12790694 0.10435312 0.06155783 0.02959793sum(wts)## [1] 1port_returns <- (sum(wts * mean_ret) + 1)^252 - 1
port_returns # we see an ___ return## [1] 0.1246907port_risk <- sqrt(t(wts) %*% (cov_mat %*% wts))
print(port_risk) # we see a ____ ## [,1]
## [1,] 0.273082# Since Risk free rate is 0%
sharpe_ratio <- port_returns/port_risk
print(sharpe_ratio) # sharpe ratio is ____## [,1]
## [1,] 0.4566052Once we have made sure the work is correct, we begin the function to run the 5000 (can use different number) trials. Firstly, create the blank vectors within the DF to store numbers.
num_port <- 15000
# Creating a matrix to store the weights
all_wts <- matrix(nrow = num_port,
ncol = length(tick))
# Creating an empty vector to store
# Portfolio returns
port_returns <- vector('numeric', length = num_port)
# Creating an empty vector to store
# Portfolio Standard deviation
port_risk <- vector('numeric', length = num_port)
# Creating an empty vector to store
# Portfolio Sharpe Ratio
sharpe_ratio <- vector('numeric', length = num_port)Now we begin the loop
for (i in seq_along(port_returns)) {
wts <- runif(length(tick))
wts <- wts/sum(wts)
# Storing weight in the matrix
all_wts[i,] <- wts
# Portfolio returns
port_ret <- sum(wts * mean_ret)
port_ret <- ((port_ret + 1)^252) - 1
# Storing Portfolio Returns values
port_returns[i] <- port_ret
# Creating and storing portfolio risk
port_sd <- sqrt(t(wts) %*% (cov_mat %*% wts))
port_risk[i] <- port_sd
# Creating and storing Portfolio Sharpe Ratios
# Assuming 0% Risk free rate
sr <- port_ret/port_sd
sharpe_ratio[i] <- sr
}Now we store the values in a table
# Storing the values in the table
portfolio_values <- tibble(Return = port_returns,
Risk = port_risk,
SharpeRatio = sharpe_ratio)
# Converting matrix to a tibble and changing column names
all_wts <- tk_tbl(all_wts)## Warning in tk_tbl.data.frame(as.data.frame(data), preserve_index, rename_index,
## : Warning: No index to preserve. Object otherwise converted to tibble
## successfully.colnames(all_wts) <- colnames(log_ret_xts)
# Combing all the values together
portfolio_values <- tk_tbl(cbind(all_wts, portfolio_values))## Warning in tk_tbl.data.frame(cbind(all_wts, portfolio_values)): Warning: No
## index to preserve. Object otherwise converted to tibble successfully.head(portfolio_values)## # A tibble: 6 × 19
## AMZN BBJP DIS EFX FCX GOOG IHI MSFT PSA PYPL
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.0165 0.110 0.104 0.0358 0.103 0.0304 0.00935 0.0899 0.0791 0.0146
## 2 0.0906 0.0401 0.0984 0.0675 0.0482 0.0426 0.0663 0.0846 0.0365 0.102
## 3 0.0116 0.112 0.142 0.0351 0.0758 0.103 0.0245 0.0942 0.0974 0.0354
## 4 0.0968 0.0159 0.0820 0.107 0.0735 0.0781 0.0789 0.0743 0.0414 0.00941
## 5 0.0459 0.0504 0.128 0.107 0.0290 0.131 0.0271 0.109 0.00607 0.0223
## 6 0.0291 0.0549 0.00565 0.0979 0.0537 0.0659 0.0539 0.110 0.0229 0.115
## # ℹ 9 more variables: RSP <dbl>, SCHG <dbl>, SPLG <dbl>, TSM <dbl>, VPU <dbl>,
## # XLRE <dbl>, Return <dbl>, Risk <dbl>, SharpeRatio <dbl>Now we will utilize our weights inputted at the top and view their statistics.
my_wts <- my_wts/sum(my_wts)
print(my_wts)## [1] 0.17125132 0.14854267 0.07737329 0.07561746 0.06063444 0.05337703
## [7] 0.04775840 0.04764134 0.04647079 0.04530025 0.04190565 0.04038394
## [13] 0.03886223 0.03839401 0.03593585 0.03055133sum(my_wts)## [1] 1my_port_returns <- (sum(my_wts * mean_ret) + 1)^252 - 1
my_port_returns # we see our return ## [1] 0.08020782my_port_risk <- sqrt(t(my_wts) %*% (cov_mat %*% my_wts))
print(my_port_risk) # we see our risk## [,1]
## [1,] 0.254466# Since Risk free rate is 0%
my_sharpe_ratio <- my_port_returns/my_port_risk
print(my_sharpe_ratio) # sharpe ratio ## [,1]
## [1,] 0.3152006my_info <- c(my_wts, my_port_returns, my_port_risk, my_sharpe_ratio)
my_labels <- c(tick, "Return", "Risk", "SharpeRatio")
my_port <- data.frame(my_labels, my_info)Then find the portfolios with min variance, optimal Sharpe ratio, and max returns
min_var <- portfolio_values[which.min(portfolio_values$Risk),]
max_sr <- portfolio_values[which.max(portfolio_values$SharpeRatio),]
max_ret <- portfolio_values[which.max(portfolio_values$Return),]Then can create some graphs and charts to see our data
Histograms that reflect holdings in the three selected portfolios
# min variance
p <- min_var %>%
gather(tick, key = Asset,
value = Weights) %>%
mutate(Asset = as.factor(Asset)) %>%
ggplot(aes(x = fct_reorder(Asset,Weights), y = Weights, fill = Asset)) +
geom_bar(stat = 'identity') +
theme_minimal() +
labs(x = 'Assets', y = 'Weights', title = "Minimum Variance Portfolio Weights") +
scale_y_continuous(labels = scales::percent) ## Warning: Using an external vector in selections was deprecated in tidyselect 1.1.0.
## ℹ Please use `all_of()` or `any_of()` instead.
## # Was:
## data %>% select(tick)
##
## # Now:
## data %>% select(all_of(tick))
##
## See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.ggplotly(p)### max returns
p <- max_ret %>%
gather(tick, key = Asset,
value = Weights) %>%
mutate(Asset = as.factor(Asset)) %>%
ggplot(aes(x = fct_reorder(Asset,Weights), y = Weights, fill = Asset)) +
geom_bar(stat = 'identity') +
theme_minimal() +
labs(x = 'Assets', y = 'Weights', title = "Maximum Returns Portfolio Weights") +
scale_y_continuous(labels = scales::percent)
ggplotly(p)### optmizied sharpe ratio
p <- max_sr %>%
gather(tick, key = Asset,
value = Weights) %>%
mutate(Asset = as.factor(Asset)) %>%
ggplot(aes(x = fct_reorder(Asset,Weights), y = Weights, fill = Asset)) +
geom_bar(stat = 'identity') +
theme_minimal() +
labs(x = 'Assets', y = 'Weights', title = "Tangency Portfolio Weights") +
scale_y_continuous(labels = scales::percent)
ggplotly(p)Then we can plot the efficient frontier and view how the portfolios fair on a complete scale
p <- portfolio_values %>%
ggplot(aes(x = Risk, y = Return, color = SharpeRatio)) +
geom_point() +
theme_classic() +
scale_y_continuous(labels = scales::percent) +
scale_x_continuous(labels = scales::percent) +
labs(x = 'Annualized Risk',
y = 'Annualized Returns',
title = "Portfolio Optimization & Efficient Frontier") +
geom_point(aes(x = Risk,
y = Return), data = min_var, color = 'red') +
geom_point(aes(x = Risk,
y = Return), data = max_sr, color = 'red')+
annotate('text', x = max_sr$Risk, y = max_sr$Return, label = "Tangency Portfolio") +
annotate('text', x = min_var$Risk + 0.025, y = min_var$Return, label = "Minimum variance portfolio") +
geom_point(aes(x = my_port_risk, y = my_port_returns), color = 'orange') +
annotate('text', x = my_port_risk, y = my_port_returns, label = "My Portfolio")
# once you find your port's risk and returns, can plot labels and point on graph
ggplotly(p)Thus we find the efficient frontier graph, be aware that this is only using past data and there is no prediction in this code.
Now that we have graphed the efficient frontier and seen how our portfolio compares to the optimized ones from 5,000 random trials. Let’s look at where we stand numerically, for some simplicity if we do not want to have to look at our random plots for each and every case.
We will look at the portfolio returns annualized, as well as it’s calculated risk, sharpe ratio, and how it compares to the benchmark. This may be more suitable for situations where we want to see how a single added holding influences these factors based on historical data.
Lets load our packages first
library(PortfolioAnalytics)## Loading required package: foreachlibrary(fPortfolio)## Loading required package: timeDate##
## Attaching package: 'timeDate'## The following objects are masked from 'package:PerformanceAnalytics':
##
## kurtosis, skewness## Loading required package: timeSeries##
## Attaching package: 'timeSeries'## The following object is masked from 'package:zoo':
##
## time<-## Loading required package: fBasics##
## Attaching package: 'fBasics'## The following object is masked from 'package:TTR':
##
## volatility## The following objects are masked from 'package:PerformanceAnalytics':
##
## kurtosis, skewness## Loading required package: fAssets# lets bind our portfolio
portfolioPrices <- NULL
for (ticker in tick) {
portfolioPrices <- cbind(portfolioPrices,
getSymbols.yahoo(ticker,
from="2020-01-01",
periodicity = 'daily', auto.assign = FALSE)[,4])
# us taking the close data
}
# check if missing data
colSums(is.na(portfolioPrices)) # checking how many NA values over the time period## SCHG.Close RSP.Close SPLG.Close IHI.Close TSM.Close MSFT.Close EFX.Close
## 0 0 0 0 0 0 0
## FCX.Close BBJP.Close XLRE.Close DIS.Close PYPL.Close AMZN.Close PSA.Close
## 0 0 0 0 0 0 0
## GOOG.Close VPU.Close
## 0 0Now we have our prices over the desired period, lets go ahead and find the returns. This gives us the portfolio returns each day as the ROC() function takes the rate of change at each close.
portfolioRets <- na.omit(ROC(portfolioPrices)) # want to omit the NA valuesFor our comparison to the benchmark we will be referring to the S&P 500, by using the SPY ticker.
benchmarkPrices <- getSymbols.yahoo('SPY', from = '2020-01-01',
periodicity = 'daily',
auto.assign = FALSE)
benchmarkRet <- na.omit(dailyReturn(benchmarkPrices$SPY.Close, type = "log"))Now we have portfolio returns and the benchmark returns, now we want to aggregate the returns using our weights to show the actual port returns
portfolioReturn <- Return.portfolio(portfolioRets)
# this is using geometric returns, base is TRUENow that we have all this stuff, we can put together some of our metrics for the portfolio like beta and alpha. This is all pretty much using the performance analytics
print(CAPM.beta(portfolioReturn, benchmarkRet, .035/252))## [1] 1.029876# the risk free rate in this case is some number over 252
# 252 represents the number of trading days in a yr
print(CAPM.jensenAlpha(portfolioReturn, benchmarkRet, 0.035/252))## [1] -0.02309981SharpeRatio(portfolioReturn, 0.035/252)## portfolio.returns
## StdDev Sharpe (Rf=0%, p=95%): 0.007061137
## VaR Sharpe (Rf=0%, p=95%): 0.004339316
## ES Sharpe (Rf=0%, p=95%): 0.002110014# we want the STDDev Sharpe
print(SharpeRatio(portfolioReturn, 0.035/252)[1,])## [1] 0.007061137Now we can show some returns annualized and calendar year
table.AnnualizedReturns(portfolioReturn)## portfolio.returns
## Annualized Return 0.0309
## Annualized Std Dev 0.2577
## Annualized Sharpe (Rf=0%) 0.1201table.CalendarReturns(portfolioReturn)## Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
## 2020 -1.1 -0.4 -1.3 -1.0 0.8 1.7 0.3 -0.4 0.6 -1.5 -0.4 0.3
## 2021 -1.8 -0.7 0.8 -0.7 0.3 -0.3 -1.2 0.2 -1.0 -0.2 -1.9 -0.3
## 2022 2.3 -0.4 -1.6 -4.4 -0.3 -1.1 1.5 -0.5 -1.1 -0.9 3.8 -0.7
## 2023 1.5 0.2 1.6 0.5 -0.7 1.6 NA NA NA NA NA NA
## portfolio.returns
## 2020 -2.4
## 2021 -6.5
## 2022 -3.6
## 2023 4.9